Feedback-controlled motion systems are mostly subject to nonlinear
friction, and the direction-dependent Coulomb friction force plays
a crucial role owing to a (theoretical) discontinuity at velocity
zero-crossings. Although the more complex dynamic friction laws
(see, e.g., [1]}, [2]}, [3]} and
references therein) allow frictional discontinuity to be bypassed
during analysis, the basic Coulomb friction phenomenon continues
to present the same challenges in terms of controller convergence,
especially in the presence of an integral control action.
Associated stick-slip behavior and so-called frictional limit
cycles were formerly addressed in [4]}. An
algebraic prediction of stick-slip, with a large set of parametric
equalities, was compared to the describing function method, while
the Coulomb plus static friction law was assumed for avoiding
discontinuity at the velocity zero-crossing. An explicit solution
for friction-generated limit cycles has also been proposed in
[5]}, necessitating static friction approximation (to
avoid discontinuity) and also requiring the stiction friction
which is larger than the Coulomb friction level. Despite including
explicit analysis of state trajectories for both sticking and
slipping phases, no straightforward conclusions about the
appearance and convergence of stick-slip behavior have been
reported. The appearance of friction-induced (so-called hunting)
limit cycles has been briefly addressed in [6]}, for
the assumed LuGre [7]} and so-called switch
[8]} friction models. Note that a former analysis of
stick-slip behavior and associated friction-induced limit cycles
can be found in [9]}. An explanation of how a
proportional-feedback-controlled motion with Coulomb friction
comes to sticking was subsequently shown in [10]} by
using the invariance principle. Stick-slip behavior, as an
observable phenomenon known in control engineering practice, was
highlighted in [1]}, and several other control
studies have since attempted to analyze and compensate such
processes. For instance, a related analysis of under-compensation
and over-compensation of static friction was reported in
[12]}. Issues associated with a slow (creeping-like)
convergence of the feedback-controlled motion in the presence of
Coulomb friction have been addressed and experimentally
demonstrated in [13]}. More recently, the
convergence problems (of PID feedback control) have been well
demonstrated, with accurate experiments reported, in
[14]} while attempting to reduce settling errors by
the reset integral control [15]}. The related analysis
has been reported before in [16]}. Despite a number
of experimental observations and elaborated studies reported in
the literature, it appears that no general consensus has been
established in relation to friction-induced stick-slip cycles in
feedback-controlled systems with Coulomb friction. In particular,
questions arise over when and under which conditions the
stick-slip cycles occur, and how a PID-controlled motion will
converge to zero equilibrium in the presence of Coulomb friction
(with discontinuity). Note that the problem of slow convergence in
vicinity to a set reference position is of particular relevance
for advanced motion control systems, see, e.g.,
[17]}, but in the PID design and tuning, see, e.g.,
[18]}, the associated issues are not widely accepted and
have yet to be formalized, that despite huge demand from precision
engineering control tasks. This gap should not come as a complete
surprise, however, given the nontrivial friction microdynamics,
visible in experimental studies
[19]}, [20]}, [21]}, and the uncertain and
time-varying friction behavior, see, e.g., [22]}.
